Question

Let $\mathbf{v}, \mathbf{w}$ be $n \times 1$ column vectors. (a) Prove that in most cases the inverse of the $n \times n$ matrix $A=\mathrm{I}-\mathbf{v} \mathbf{w}^T$ has the form $A^{-1}=\mathrm{I}-c \mathbf{v} \mathbf{w}^T$ for some scalar $c$. Find all $\mathbf{v}, \mathbf{w}$ for which such a result is valid. (b) Illustrate the method when $\mathbf{v}=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{r}-1 \\ 2\end{array}\right)$. (c) What happens when the method fails?

    Let $\mathbf{v}, \mathbf{w}$ be $n \times 1$ column vectors. (a) Prove that in most cases the inverse of the $n \times n$ matrix $A=\mathrm{I}-\mathbf{v} \mathbf{w}^T$ has the form $A^{-1}=\mathrm{I}-c \mathbf{v} \mathbf{w}^T$ for some scalar $c$. Find all $\mathbf{v}, \mathbf{w}$ for which such a result is valid. (b) Illustrate the method when $\mathbf{v}=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{r}-1 \\ 2\end{array}\right)$. (c) What happens when the method fails?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 1, Problem 16

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We need to find conditions under which $A^{-1}$ has the form $\mathrm{I} - c \mathbf{v} \mathbf{w}^T$ for some scalar $c$. Show more…

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Let $\mathbf{v}, \mathbf{w}$ be $n \times 1$ column vectors. (a) Prove that in most cases the inverse of the $n \times n$ matrix $A=\mathrm{I}-\mathbf{v} \mathbf{w}^T$ has the form $A^{-1}=\mathrm{I}-c \mathbf{v} \mathbf{w}^T$ for some scalar $c$. Find all $\mathbf{v}, \mathbf{w}$ for which such a result is valid. (b) Illustrate the method when $\mathbf{v}=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{r}-1 \\ 2\end{array}\right)$. (c) What happens when the method fails?

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